How to prove that tan112 1/2=-sqrt2-1?

2 Answers
May 30, 2018

Please see the proof below

Explanation:

We need

cos(2x)=2cos^2x-1=1-2sin^2x

Therefore,

cos(x/2)=sqrt((1+cosx)/2)

sin(x/2)=sqrt((1-cosx)/2)

tan(x/2)=sin(x/2)/cos(x/2)=sqrt((1-cosx)/(1+cosx))

=sqrt(((1-cosx)(1-cosx))/((1+cosx)(1-cosx)))

=sqrt((1-cosx)^2/(1-cos^2x))

=sqrt((1-cosx)^2/(sin^2x))

And finally,

tan(x/2)=(1-cosx)/sinx

Here,

x=225

cos(225)=-1/sqrt2

sin(225)=-1/sqrt2

tan(225/2)=(1-(-1/sqrt2))/(-1/sqrt2)=-(sqrt2+1)

May 30, 2018

"see explanation"

Explanation:

"using the "color(blue)"half angle identity"

•color(white)(x)tan(x/2)=+-sqrt((1-cosx)/(1+cosx))

112 1/2" is in the second quadrant where"

tan(112 1/2)<0

tan(112 1/2)=-sqrt((1-cos225)/(1+cos225))

color(white)(xxxxxxxx)=-sqrt((1-(-cos45))/(1+(-cos45))

color(white)(xxxxxxxx)=-sqrt((1+1/sqrt2)/(1-1/sqrt2))

color(white)(xxxxxxxx)=-sqrt((sqrt2+1)/(sqrt2-1))

color(white)(xxxxxxxx)=-sqrt((sqrt2+1)^2/((sqrt2-1)(sqrt2+1))

color(white)(xxxxxxxx)=-sqrt((sqrt2+1)^2)

color(white)(xxxxxxxx)=-(sqrt2+1)=-sqrt2-1